Inclined scanning, auxiliary tables

Kopskyacute, V.; Litvin, D. B.

doi:10.1107/97809553602060000652

International
Tables for
Crystallography
Volume E
Subperiodic groups
Edited by V. Kopský and D. B. Litvin

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International Tables for Crystallography (2006). Vol. E. ch. 5.2, pp. 403-404 | 1 | 2 |

Section 5.2.4.3.2. Inclined scanning, auxiliary tables

V. Kopský^a ^* and D. B. Litvin^b

^a Department of Physics, University of the South Pacific, Suva, Fiji, and Institute of Physics, The Academy of Sciences of the Czech Republic, Na Slovance 2, PO Box 24, 180 40 Prague 8, Czech Republic, and ^bDepartment of Physics, Penn State Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 19610–6009, USA
Correspondence e-mail: kopsky@fzu.cz

5.2.4.3.2. Inclined scanning, auxiliary tables

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Orientation orbits ( [mn0] ), ( [0mn] ) and ( [n0m] ): Each of these orientations has a scanning group of monoclinic symmetry, namely: 2 ( $[C_{2}]$ ) for space groups of the geometric class [222] ( $[D_{2}]$ ) in all settings and for the groups of the class [mm2] ( $[C_{2v}]$ ) in the ( [abc] ) setting; m ( $[C_{s}]$ ) for groups of the geometric class [mm2] ( $[C_{2v}]$ ) in the settings ( [bac] ), ( $[c{\bar a}b]$ ); and [2/m] ( $[C_{2h}]$ ) for groups of the geometric class [mmm] ( $[D_{2h}]$ ) in all settings. In each case, the scanning group $[\cal H]$ is a halving subgroup of the scanned group $[\cal G]$ and consequently each orientation orbit contains two orientations with the same scanning group. The symmetry increases to orthorhombic and the orbit contains one orientation for the special values $[m = \pm 1]$ , [n = 0] or [m = 0] , $[n = \pm 1]$ .

The scanning groups are monoclinic and one can deduce them by viewing the diagrams of the scanned groups. By omitting the axes parallel with and planes perpendicular to the diagram plane, the diagram of the scanning group remains. This, however, is the diagram of the monoclinic scanning group in its standard setting when the unique axis is perpendicular to the plane of the diagram. This unique axis is a common direction for both orientations of the orbit and it is the c axis for the orientation orbit ( [mn0] ), and the a axis and b axis for the orientation orbits ( [0mn] ) and ( [n0m] ), respectively. The basis $[{\bf a}']$ , $[{\bf b}']$ , $[{\bf d}]$ for the scanning group is expressed in the same way through the auxiliary basis and the scanning groups in the reference tables are given by their Hermann–Mauguin symbols with reference to the auxiliary bases. The three orbits are separated by horizontal lines in the tables of orientation orbits as well as in the first column of the reference tables. If the scanning group in a column for a certain scanned group $[\cal G]$ has the same symbol for orbits in adjoining rows, we give it in a block common to these orbits.

References

International Tables for Crystallography (2006). Vol. E. ch. 5.2, pp. 403-404